nGeneHS — Hemodynamic Simulation for de Laval Nozzle

nGeneHS_deLavalNozzle: Class inheritance for nozzle model

Equation \( V = f(x) \)

Determine gas velocity \( V \) as a function of axial distance \( x \) from the throat (\( x = 0 \)).

(1) Nozzle Geometry \( A(x) \)

• Converging Section (\( x < 0 \)):

  \( A(x) = A^* \bigl(1 - kx\bigr) \quad\text{for}\; x \in [-L_{\text{converging}},\,0] \)

  \( k \) is a positive constant controlling how quickly the area decreases as \( x \) approaches the throat from the chamber.

• Throat (\( x = 0 \)):

  \( A(0) = A^* \)

• Diverging Section (\( x > 0 \)):

  \( A(x) = A^* \bigl(1 + k' x^2\bigr) \quad\text{for}\; x \in [0,\,L_{\text{diverging}}] \)

  \( k' \) is a positive constant defining the expansion rate in the diverging section; the quadratic term yields a smooth, realistic area growth.

(2) Area–Velocity Relationship

The area–velocity relation in a de Laval nozzle follows compressible-flow dynamics:

\( V(x) = M(x)\,\gamma\,R\,T(x) \)

where \( M(x) \) is the local Mach number, obtained from

\( M(x) = \sqrt{\frac{2}{\gamma-1}\Bigl[\Bigl(\tfrac{A(x)}{A^*}\Bigr)^{\!\tfrac{2}{\gamma-1}}\Bigl(1-\bigl(\tfrac{p_e}{p_0}\bigr)^{\!\tfrac{\gamma-1}{\gamma}}\Bigr)\Bigr]} \)

The static temperature is

\( T(x) = T_0\Bigl(1 + \frac{\gamma - 1}{2} M(x)^2\Bigr)^{-1} \)

Combining,

\( T(x) = T_0\Bigl(1 + \frac{\gamma - 1}{2} \Bigl[\sqrt{\frac{2}{\gamma-1}\Bigl(\tfrac{A(x)}{A^*}\Bigr)^{\!\tfrac{2}{\gamma-1}}\Bigl(1-\bigl(\tfrac{p_e}{p_0}\bigr)^{\!\tfrac{\gamma-1}{\gamma}}\Bigr)}\Bigr]^2\Bigr)^{-1} \)